A reduced FVE formulation based on POD method and error analysis for two-dimensional viscoelastic problem

Proper orthogonal decomposition (POD) method has been successfully used in the reduced-order modeling of complex systems. In this paper, we extend the applications of POD method, i.e., combine a classical finite volume element (FVE) method with POD method to establish a reduced FVE formulation with lower dimensions and sufficiently high accuracy for two-dimensional viscoelastic problem with real practical applied background, and analyze the errors between the reduced POD FVE solution and the classical FVE solution so as to provide scientific theoretic basis for service applications. Some numerical examples illustrate the fact that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is also shown that the reduced FVE formulation based on POD method is feasible and efficient for solving two-dimensional viscoelastic problem.     

A stream function-vorticity formulation coupled with boundary integrals for the two-dimensional exterior Stokes problem

In this work, we derive a stream function-vorticity variational formulation coupled with boundary integrals for the exterior Stokes problem in two dimensions, when the right-hand side has a bounded support. The stream function-vorticity formulation is expressed in a bounded region containing the support of the right-hand side, and the boundary conditions on the artificial boundary are obtained by an integral representation. We prove that this coupled formulation is equivalent to the original Stokes problem.     

CABARET scheme in velocity-pressure formulation for two-dimensional incompressible fluids

The CABARET method was generalized to two-dimensional incompressible fluids in terms of velocity and pressure. The resulting algorithm was verified by computing the transport and interaction of various vortex structures: a stationary and a moving solitary vortex, Taylor-Green vortices, and vortices formed by the instability of double shear layers. Much attention was also given to the modeling of homogeneous isotropic turbulence and to the analysis of its spectral properties. It was shown that, regardless of the mesh size, the slope of the energy spectra up to the highest-frequency harmonics is equal ?3, which agrees with Batchelor’s enstrophy cascade theory.     

A reduced finite element formulation based on POD method for two-dimensional solute transport problems

A proper orthogonal decomposition (POD) method is applied to a usual finite element (FE) formulation for two-dimensional solute transport problems with real practical applied background such that it is reduced into a reduced FE formulation with lower dimensions and high enough accuracy. The error estimates between the reduced POD FE solutions and the usual FE solutions are provided. It is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is also shown that this validates the feasibility and efficiency of POD FE method.     

A two-dimensional problem for an anisotropic body with a finite number of “tunnel” cracks

We give the results of studies of the stress state for an infinite anisotropic body with a number of planar cracks along a single plane. For the simplest types of load we prove that the stress intensity factors is independent of the type of anisotropy. We describe the results of numerical studies as functions of the geometric characteristics of the body. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 27–34     

A comparative numerical analysis for the guillotine two-dimensional cutting problem

In this work, the behavior of four algorithms in the resolution of the two-dimensional constrained guillotine cutting problem is analyzed. This problem is concerned about the way a set of pieces should be cut from a plate of greater dimensions, considering guillotine cutting and a constrained number of times a piece can be cut from the plate. In this study three combinatorial and two heuristic methods are considered. In the combinatorial methods from the set of pieces, a minimum loss layout is constructively generated based on Wang's algorithm. In addition, an evolutionary and an annealing type approach are considered. All of these models have been implemented on a high performance Silicon Graphics machine. Performance of each algorithm is analyzed both in terms of percentage waste and running time. In order to do that, a set of 1000 instances are classified according to their combinatorial degree and subsequently evaluated. This revised version was published online in June 2006 with corrections to the Cover Date.     

A PDAE formulation of parabolic problems with dynamic boundary conditions

The weak formulation of parabolic problems with dynamic boundary conditions is rewritten in form of a partial differential–algebraic equation. More precisely, we consider two dynamic equations with a coupling condition on the boundary. This constraint is included explicitly as an additional equation and incorporated with the help of a Lagrange multiplier. Well-posedness of the formulation is shown.     

A reduced-order extrapolation algorithm based on CNLSMFE formulation and POD technique for two-dimensional Sobolev equations

A reduced-order extrapolation algorithm based on Crank-Nicolson least-squares mixed finite element （CNLSMFE） formulation and proper orthogonal decomposition （POD） technique for two-dimensional （2D） Sobolev equations is established. The error estimates of the reduced-order CNLSMFE solutions and the implementation for the reduced-order extrapolation algorithm are provided. A numerical example is used to show that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order extrapolation algorithm is feasible and efficient for seeking numerical solutions to 2D Sobolev equations.     

Effective elastic moduli of two-dimensional solids with multiple cracks

An accurate method which directly accounts for the interactions between different microcracks is used for analyzing the elastic problem of multiple cracks solids. The effective elastic moduli for randomly oriented cracks and parallel cracks are evaluated for the representative volume element (RVE) with microcracks in infinite media. The numerical results are compared with those from various micromechanics models and experimental data. These results show that the present method is simple and provides a direct and efficient approach to dealing with elastic solids containing multiple cracks. Project supported by the National Natural Science Foundation of China (Grant No. 19704100), and the National Natural Science Foundation of Chinese Academy of Sciences (Project KJ951-1-201).     

Theory of photorefractive dynamic grating formulation

The photorefractive holographic recording and two-beam coupling are both dynamic grating formulation process. The interference light intensity of the two coherent beams induces a phase grating though photo-induced refractive index variation and the phase grating changing the intensities of the two beams through beam-coupling take place at the same time. By solving simultaneously the band transport equations and wave-coupled equations, and using the light intensity modulation as the main variable, the analytic solution is obtained, which is valid for any light intensity modulation and constant light excitation efficiency. Here all the mechanics of drift, diffusion and photovoltaic effect are considered. The result shows that the modulation of the dynamic grating varies more slowly compared with that of the linear modulation approximation.     

A new boundary-type scheme for sensitivity analysis using Trefftz formulation

This paper presents a new boundary-type scheme for a sensitivity analysis of the two-dimensional potential problem by using the Trefftz formulation.

Since the Trefftz method is the boundary-type solution procedure, input data generation is easier than the domain-type solution procedure. Moreover, the physical quantities are expressed by the regular equations, their sensitivities, which is derived from the direct differentiation of the original quantaties, are also regular. Therefore, they can be calculated more easily than the ordinary boundary element method using the singular boundary integral equation. The present schemes are applied to simple numerical examples in order to confirm the validity of the present formulation.     

On the fundamental relations of the two-dimensional theory of cracks of an anisotropic body

Representations obtained by the author in an earlier paper for the complex potentials of the general two-dimensional problem of the theory of cracks of an anisotropic body are brought into more convenient form. A new form is introduced for the stress intensity factors. For certain types of load the value of the stress intensity factors are exhibited. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 18–26     

Use of the elastic-and-rigid-combined beam element for dynamic analysis of a two-dimensional frame with arbitrarily distributed rigid beam segments

This paper presents an elastic-and-rigid-combined beam element such that the dynamic characteristics of a hybrid beam and a two-dimensional frame composed of any number of elastic and rigid beam segments can be easily determined. First of all, the displacements for the two nodes of a rigid beam segment are determined in terms of the displacements of its centre of gravity (c.g.). Next, the mass and stiffness matrices for the elastic-and-rigid-combined beam element are derived using the above-mentioned nodal displacements of the rigid beam segment and those of the two adjacent elastic beam elements. Furthermore, for the transformation of state variables of last elastic-and-rigid-combined (or three-node) beam element between the local and global co-ordinate systems, a new transformation matrix is also presented. Finally, the overall property matrices of the entire vibrating system are determined with the conventional assembly technique of finite element method (FEM) and its natural frequencies and associated mode shapes are determined with the standard approach. Some important factors, such as length of rigid beam segment, position for the centre of gravity (c.g.) of rigid beam segment, and total number of rigid beam segments in the entire vibrating system, are investigated. Numerical results reveal that the above-mentioned parameters have significant influence on the dynamic characteristics of the structure with arbitrarily distributed rigid beam segments.     

A compact formulation of an elastoplastic analysis problem

Two important problems in the area of engineering plasticity are limit load analysis and elastoplastic analysis. It is well known that these two problems can be formulated as linear and quadratic programming problems, respectively (Refs. 1–2). In applications, the number of variables in each of these mathematical programming problems tends to be large. Consequently, it is important to have efficient numerical methods for their solution. The purpose of this paper is to present a method which allows the quadratic programming formulation of the elastoplastic analysis to be reformulated as an equivalent quadratic programming problem which has significantly fewer variables than the original formulation. Indeed, in Section 4, we will present details of an example for which the original quadratic programming formulation required 297 variables and for which the equivalent formulation presented here required only two variables. The method is based on a characterization of the entire family of optimal solutions for a linear programming problem.This research was supported by the Natural Science and Engineering Council of Canada under Grant No. A8189 and by a Leave Fellowship from the Social Sciences and Humanities Research Council of Canada. The author takes pleasure in acknowledging many stimulating discussions with Professor D. E. Grierson.     

A general formulation for mean-value analysis in product-form batch-movement queueing networks

A number of recent papers have shown that many classes of queueing networks with batches of customers served and routed through the network have equilibrium distributions which factorise into product forms over the nodes of the network. In this paper we demonstrate how such networks are amenable to a mean-value analysis which generalises that used for single-movement networks.Since product-form stochastic Petri nets (SPNs) can be viewed as batch-movement queueing networks, our algorithm is also applicable to their analysis.     

Complete system of two-dimensional invariants for formulation of triangular finite element of composite shells

The paper describes a system of invariants of symmetric two-dimensional tensors defined on a plane or a surface. The system comprises the well-known first and second invariants and a new quantity called the combined invariant of two tensors. The focus is on the expression for the invariants in terms of normal components of the tensors determined in three different directions on the surface. The system of invariants is used to construct a triangular finite element for geometrically nonlinear analysis of shear deformable anisotropic shells subject to the Reissner–Mindlin assumptions. The relations obtained allow one to readily determine the strain energy of the element for the normal components of the stress and strain tensors in the direction of the element edges. Numerical examples are given to demonstrate some nonlinear capabilities of the element.